Theorem bnj154 35374

Description: Technical lemma for bnj153 35376. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)

Hypotheses

Ref	Expression
bnj154.1	⊢ (𝜑1 ↔ [𝑔 / 𝑓]𝜑′)
bnj154.2	⊢ (𝜑′ ↔ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅))

Assertion

Ref	Expression
bnj154	⊢ (𝜑1 ↔ (𝑔‘∅) = pred(𝑥, 𝐴, 𝑅))

Distinct variable groups:   𝐴,𝑓   𝑅,𝑓   𝑓,𝑔   𝑥,𝑓

Allowed substitution hints:   𝐴(𝑥, 𝑔)   𝑅(𝑥, 𝑔)   𝜑′(𝑥, 𝑓, 𝑔)   𝜑₁(𝑥, 𝑓, 𝑔)

Proof of Theorem bnj154

Step	Hyp	Ref	Expression
1		bnj154.1	. 2 ⊢ (𝜑1 ↔ [𝑔 / 𝑓]𝜑′)
2		bnj154.2	. . 3 ⊢ (𝜑′ ↔ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅))
3	2	sbcbii 3798	. 2 ⊢ ([𝑔 / 𝑓]𝜑′ ↔ [𝑔 / 𝑓](𝑓‘∅) = pred(𝑥, 𝐴, 𝑅))
4		vex 3457	. . 3 ⊢ 𝑔 ∈ V
5		fveq1 6881	. . . 4 ⊢ (𝑓 = 𝑔 → (𝑓‘∅) = (𝑔‘∅))
6	5	eqeq1d 2764	. . 3 ⊢ (𝑓 = 𝑔 → ((𝑓‘∅) = pred(𝑥, 𝐴, 𝑅) ↔ (𝑔‘∅) = pred(𝑥, 𝐴, 𝑅)))
7	4, 6	sbcie 3783	. 2 ⊢ ([𝑔 / 𝑓](𝑓‘∅) = pred(𝑥, 𝐴, 𝑅) ↔ (𝑔‘∅) = pred(𝑥, 𝐴, 𝑅))
8	1, 3, 7	3bitri 300	1 ⊢ (𝜑1 ↔ (𝑔‘∅) = pred(𝑥, 𝐴, 𝑅))

This proof depends on syntax axioms:   ↔ wb 209   = wceq 1570  [wsbc 3742  ∅c0 4282  ‘cfv 6537   predc-bnj14 35185

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2734

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-ex 1813  df-sb 2100  df-clab 2741  df-cleq 2754  df-clel 2837  df-v 3455  df-sbc 3743  df-ss 3919  df-uni 4871  df-br 5108  df-iota 6493  df-fv 6545

This theorem is used by:  bnj153  35376  bnj580  35409  bnj607  35412