Theorem bnj526 35046

Description: Technical lemma for bnj852 35079. 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
bnj526.1	⊢ (𝜑 ↔ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅))
bnj526.2	⊢ (𝜑″ ↔ [𝐺 / 𝑓]𝜑)
bnj526.3	⊢ 𝐺 ∈ V

Assertion

Ref	Expression
bnj526	⊢ (𝜑″ ↔ (𝐺‘∅) = pred(𝑋, 𝐴, 𝑅))

Distinct variable groups:   𝐴,𝑓   𝑓,𝐺   𝑅,𝑓   𝑓,𝑋

Allowed substitution hints:   𝜑(𝑓)   𝜑″(𝑓)

Proof of Theorem bnj526

Step	Hyp	Ref	Expression
1		bnj526.2	. 2 ⊢ (𝜑″ ↔ [𝐺 / 𝑓]𝜑)
2		bnj526.1	. . 3 ⊢ (𝜑 ↔ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅))
3	2	sbcbii 3798	. 2 ⊢ ([𝐺 / 𝑓]𝜑 ↔ [𝐺 / 𝑓](𝑓‘∅) = pred(𝑋, 𝐴, 𝑅))
4		bnj526.3	. . 3 ⊢ 𝐺 ∈ V
5		fveq1 6834	. . . 4 ⊢ (𝑓 = 𝐺 → (𝑓‘∅) = (𝐺‘∅))
6	5	eqeq1d 2739	. . 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 297	1 ⊢ (𝜑″ ↔ (𝐺‘∅) = pred(𝑋, 𝐴, 𝑅))

Syntax hints:   ↔ wb 206   = wceq 1542   ∈ wcel 2114  Vcvv 3441  [wsbc 3741  ∅c0 4286  ‘cfv 6493   predc-bnj14 34846

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1545  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-v 3443  df-sbc 3742  df-ss 3919  df-uni 4865  df-br 5100  df-iota 6449  df-fv 6501

This theorem is referenced by:  bnj607  35074