Theorem bnj523 32867

Description: Technical lemma for bnj852 32901. 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
bnj523.1	⊢ (𝜑 ↔ (𝐹‘∅) = pred(𝑋, 𝐴, 𝑅))
bnj523.2	⊢ (𝜑′ ↔ [𝑀 / 𝑛]𝜑)
bnj523.3	⊢ 𝑀 ∈ V

Assertion

Ref	Expression
bnj523	⊢ (𝜑′ ↔ (𝐹‘∅) = pred(𝑋, 𝐴, 𝑅))

Distinct variable groups:   𝐴,𝑛   𝑛,𝐹   𝑅,𝑛   𝑛,𝑋

Allowed substitution hints:   𝜑(𝑛)   𝑀(𝑛)   𝜑′(𝑛)

Proof of Theorem bnj523

Step	Hyp	Ref	Expression
1		bnj523.2	. 2 ⊢ (𝜑′ ↔ [𝑀 / 𝑛]𝜑)
2		bnj523.1	. . 3 ⊢ (𝜑 ↔ (𝐹‘∅) = pred(𝑋, 𝐴, 𝑅))
3	2	sbcbii 3776	. 2 ⊢ ([𝑀 / 𝑛]𝜑 ↔ [𝑀 / 𝑛](𝐹‘∅) = pred(𝑋, 𝐴, 𝑅))
4		bnj523.3	. . 3 ⊢ 𝑀 ∈ V
5	4	bnj525 32718	. 2 ⊢ ([𝑀 / 𝑛](𝐹‘∅) = pred(𝑋, 𝐴, 𝑅) ↔ (𝐹‘∅) = pred(𝑋, 𝐴, 𝑅))
6	1, 3, 5	3bitri 297	1 ⊢ (𝜑′ ↔ (𝐹‘∅) = pred(𝑋, 𝐴, 𝑅))

Syntax hints:   ↔ wb 205   = wceq 1539   ∈ wcel 2106  Vcvv 3432  [wsbc 3716  ∅c0 4256  ‘cfv 6433   predc-bnj14 32667

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1542  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-sbc 3717

This theorem is referenced by:  bnj600  32899  bnj908  32911  bnj934  32915