Theorem bnj154 34725

Description: Technical lemma for bnj153 34727. 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 3837	. 2 ⊢ ([𝑔 / 𝑓]𝜑′ ↔ [𝑔 / 𝑓](𝑓‘∅) = pred(𝑥, 𝐴, 𝑅))
4		vex 3466	. . 3 ⊢ 𝑔 ∈ V
5		fveq1 6902	. . . 4 ⊢ (𝑓 = 𝑔 → (𝑓‘∅) = (𝑔‘∅))
6	5	eqeq1d 2728	. . 3 ⊢ (𝑓 = 𝑔 → ((𝑓‘∅) = pred(𝑥, 𝐴, 𝑅) ↔ (𝑔‘∅) = pred(𝑥, 𝐴, 𝑅)))
7	4, 6	sbcie 3820	. 2 ⊢ ([𝑔 / 𝑓](𝑓‘∅) = pred(𝑥, 𝐴, 𝑅) ↔ (𝑔‘∅) = pred(𝑥, 𝐴, 𝑅))
8	1, 3, 7	3bitri 296	1 ⊢ (𝜑1 ↔ (𝑔‘∅) = pred(𝑥, 𝐴, 𝑅))

Syntax hints:   ↔ wb 205   = wceq 1534  [wsbc 3776  ∅c0 4325  ‘cfv 6556   predc-bnj14 34535

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2697

This theorem depends on definitions:  df-bi 206  df-an 395  df-tru 1537  df-ex 1775  df-sb 2061  df-clab 2704  df-cleq 2718  df-clel 2803  df-v 3464  df-sbc 3777  df-ss 3964  df-uni 4916  df-br 5156  df-iota 6508  df-fv 6564

This theorem is referenced by:  bnj153  34727  bnj580  34760  bnj607  34763