Theorem bnj206 32759

Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)

Hypotheses

Ref	Expression
bnj206.1	⊢ (𝜑′ ↔ [𝑀 / 𝑛]𝜑)
bnj206.2	⊢ (𝜓′ ↔ [𝑀 / 𝑛]𝜓)
bnj206.3	⊢ (𝜒′ ↔ [𝑀 / 𝑛]𝜒)
bnj206.4	⊢ 𝑀 ∈ V

Assertion

Ref	Expression
bnj206	⊢ ([𝑀 / 𝑛](𝜑 ∧ 𝜓 ∧ 𝜒) ↔ (𝜑′ ∧ 𝜓′ ∧ 𝜒′))

Proof of Theorem bnj206

Step	Hyp	Ref	Expression
1		sbc3an 3791	. 2 ⊢ ([𝑀 / 𝑛](𝜑 ∧ 𝜓 ∧ 𝜒) ↔ ([𝑀 / 𝑛]𝜑 ∧ [𝑀 / 𝑛]𝜓 ∧ [𝑀 / 𝑛]𝜒))
2		bnj206.1	. . . 4 ⊢ (𝜑′ ↔ [𝑀 / 𝑛]𝜑)
3	2	bicomi 223	. . 3 ⊢ ([𝑀 / 𝑛]𝜑 ↔ 𝜑′)
4		bnj206.2	. . . 4 ⊢ (𝜓′ ↔ [𝑀 / 𝑛]𝜓)
5	4	bicomi 223	. . 3 ⊢ ([𝑀 / 𝑛]𝜓 ↔ 𝜓′)
6		bnj206.3	. . . 4 ⊢ (𝜒′ ↔ [𝑀 / 𝑛]𝜒)
7	6	bicomi 223	. . 3 ⊢ ([𝑀 / 𝑛]𝜒 ↔ 𝜒′)
8	3, 5, 7	3anbi123i 1155	. 2 ⊢ (([𝑀 / 𝑛]𝜑 ∧ [𝑀 / 𝑛]𝜓 ∧ [𝑀 / 𝑛]𝜒) ↔ (𝜑′ ∧ 𝜓′ ∧ 𝜒′))
9	1, 8	bitri 275	1 ⊢ ([𝑀 / 𝑛](𝜑 ∧ 𝜓 ∧ 𝜒) ↔ (𝜑′ ∧ 𝜓′ ∧ 𝜒′))

Syntax hints:   ↔ wb 205   ∧ w3a 1087   ∈ wcel 2104  Vcvv 3437  [wsbc 3721

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-ext 2707

This theorem depends on definitions:  df-bi 206  df-an 398  df-3an 1089  df-tru 1542  df-ex 1780  df-sb 2066  df-clab 2714  df-cleq 2728  df-clel 2814  df-v 3439  df-sbc 3722

This theorem is referenced by:  bnj124  32900  bnj207  32910